If $f_{1}(x,y)$ and $f_{2}(x,y)$ are two homogeneous polynomials then prove that :
1
$\frac{f_{1}(x,y)+f_{2}(x,y)}{xf_{1}(x,y)+yf_{2}(x,y)}$ is homogeneous.
2
Can i replace the denominator part by anyother homogeneous polynomial? Will it be homogeneous then ? Say $f_{3}(x,y)$ be another homogeneous polynomials then what about $\frac{f_{1}(x,y)+f_{2}(x,y)}{f_{3}(x,y)}$ ?
A function $h( \mathbf x)$, where $\mathbf x=(x_1,x_2,...,x_n)$ is said to be homogeneous of degree $k$ if
$$h( \lambda \mathbf x)=\lambda^k h( \mathbf x)$$
for all $\lambda \in \Bbb R.$
Even though $f_1(x,y)$ and $f_2(x,y)$ are both homogeneous, their sum will, in general, not be homogeneous, because their degree of homogeneity need not be the same
$$f_1(\lambda x, \lambda y)+f_2(\lambda x,\lambda y)=\lambda^{k_1} f_1(x,y)+\lambda^{k_2} f_2(x,y) \neq \lambda^k[f_1(x,y)+f_2(x,y)].$$
Of course, if it happens that their degree of homogeneity is the same, then $k_1=k_2=k$ and
$$f_1(\lambda x, \lambda y)+f_2(\lambda x,\lambda y)=\lambda^{k} f_1(x,y)+\lambda^{k} f_2(x,y)=\lambda^{k} [f_1(x,y)+f_2(x,y)].$$
Therefore,
$\frac{f_{1}(x,y)+f_{2}(x,y)}{xf_{1}(x,y)+yf_{2}(x,y)}$ is homogeneous if the degree of homogeneity of $f_{i}(x,y)$ is the same for all $i$.
And you can replace the denominator part by any other homogeneous polynomial. In this case, the degree of homogeneity does not matter, because (assuming again that $f_1(x,y)$ and $f_2(x,y)$ are both homogeneous), we have that
$$\frac{f_1(\lambda x, \lambda y)}{f_2(\lambda x,\lambda y)}=\frac{\lambda^{k_1} f_1(x,y)}{\lambda^{k_2} f_2(x,y)}=\lambda^{k_1-k_2}\frac{f_1(x,y)}{f_2(x,y)}.$$